A275480 Decimal expansion of B, a constant appearing in an asymptotic formula related to the exponential divisor function sigma^(e).

5, 6, 8, 2, 8, 5, 4, 9, 3, 7, 4, 6, 8, 0, 6, 9, 5, 4, 3, 2, 6, 0, 3, 6, 5, 7, 6, 1, 9, 1, 9, 1, 6, 2, 9, 6, 7, 2, 4, 4, 0, 4, 5, 0, 9, 3, 1, 9, 7, 8, 6, 3, 8, 3, 9, 4, 5, 2, 6, 3, 2, 7, 2, 1, 5, 8, 1, 1, 9, 8, 6, 0, 1, 5, 7, 5, 7, 6, 4, 4, 1, 8, 4, 3, 8, 0, 6, 9, 6, 3, 6, 3, 7, 4, 4, 9, 2, 7, 6, 3, 1, 0, 9, 6, 1, 2

Offset: 0

Author

Jean-François Alcover, Jul 29 2016

Keywords

• nonn
• cons

Examples

			0.5682854937...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.11 Abundant numbers density constant p. 126.

Links

• Eric Weisstein's World of Mathematics, e-Divisor

Crossrefs

Cf. A072691 (value of the same limit sum with sigma(n) instead of sigma^(e)(n)).

Programs

Mathematica

digits = 10; maxPi = 10^5;
B = (1/2)*Product[1 + 1/(p*(p^2-1)) - 1/(p^2-1) + (1-1/p)*((Log[-(1/p^2)] - Log[1/p^2] + QPolyGamma[0, -(Log[-(1/p^2)]/Log[p]), p] - QPolyGamma[0, -(Log[1/p^2]/Log[p]), p])/(2*Log[p])), {p,Prime[Range[maxPi]]}];
RealDigits[N[B] // Chop, 10, digits][[1]]
$MaxExtraPrecision = 2000; Do[m = 2000; Clear[f]; f[p_] := (1 + 1/(p*(p^2 - 1)) - 1/(p^2 - 1) + (1 - 1/p)*Sum[p^k/(p^(2 k) - 1), {k, 2, kmax}]); c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2]/2 * Exp[N[Sum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 112]]], {kmax, 100, 1000, 100}] (* Vaclav Kotesovec, Jun 19 2020 *)

Formula

B = lim_{N->inf} (1/N^2) * Sum_{n=1..N} sigma^(e)(n), where sigma^(e)(n) is the sum of all exponential divisors of n.

B = (1/2) * Product_{p prime} (1 + 1/(p*(p^2 - 1)) - 1/(p^2 - 1) + (1 - 1/p)*Sum_{k>=2} p^k/(p^(2k)-1)).

Extensions

More digits from Robert G. Wilson v, Feb 25 2019

More digits from Vaclav Kotesovec, Jun 19 2020